Free Access
Issue
Math. Model. Nat. Phenom.
Volume 8, Number 2, 2013
Anomalous diffusion
Page(s) 17 - 27
DOI https://doi.org/10.1051/mmnp/20138202
Published online 24 April 2013
  1. A. D. Fokker. Die mittlere energie rotierender elektrischer dipole im strahlungsfeld. Annalen der Physik, vol. 348, no. 5, (1914), pp. 810-820. [CrossRef] [Google Scholar]
  2. M. Planck, Sitzber. Preu. Akad. Wiss., (1917), p. 324. [Google Scholar]
  3. H. Risken. The Fokker-Planck equation: Methods of solution and applications. Second Edition., vol. 18. Springer Verlag, 1996. [Google Scholar]
  4. E. Barkai, R. Metzler, J. Klafter. From continuous time random walks to the fractional Fokker-Planck equation. Phys. Rev. E, vol. 61, no. 1, (2000), p. 132. [CrossRef] [MathSciNet] [Google Scholar]
  5. I. M. Sokolov, J. Klafter. Field-induced dispersion in subdiffusion. Phys. Rev. Lett., vol. 97, no. 14, (2006), p. 140602. [CrossRef] [PubMed] [Google Scholar]
  6. M. Magdziarz, A. Weron, J. Klafter. Equivalence of the fractional Fokker-Planck and subordinated Langevin equations: The case of a time-dependent force. Phys. Rev. Lett., vol. 101, no. 21, (2008), p. 210601. [CrossRef] [PubMed] [Google Scholar]
  7. B. I. Henry, T. A. M. Langlands, P. Straka. Fractional Fokker-Planck equations for subdiffusion with space- and time-dependent forces. Phys. Rev. Lett., vol. 105, no. 17, (2010), p. 170602. [CrossRef] [PubMed] [Google Scholar]
  8. A. Weron, M. Magdziarz, K. Weron. Modeling of subdiffusion in space-time-dependent force fields beyond the fractional Fokker-Planck equation. Phys. Rev. E, vol. 77, no. 3, (2008), p. 036704. [CrossRef] [Google Scholar]
  9. M. G. Hahn, K. Kobayashi, S. Umarov. Fokker-Planck-Kolmogorov equations associated with time-changed fractional Brownian motion. Proc. Amer. Math. Soc., (2011), pp. 691-705. [Google Scholar]
  10. V. P. Shkilev. Subdiffusion in a time-dependent force field. J. Exp. Theor. Phys., vol. 114, (2012), p. 830. [CrossRef] [Google Scholar]
  11. B. I. Henry, S. L. Wearne. Fractional reaction-diffusion. Physica A, vol. 276, no. 3, (2000), pp. 448-455. [CrossRef] [Google Scholar]
  12. B. I. Henry, T. A. M. Langlands, S. L. Wearne. Anomalous diffusion with linear reaction dynamics: From continuous time random walks to fractional reaction-diffusion equations. Phys. Rev. E, vol. 74, no. 3, (2006), p. 031116. [CrossRef] [MathSciNet] [Google Scholar]
  13. I. M. Sokolov, M. G. W. Schmidt, F. Sagués. Reaction-subdiffusion equations. Phys. Rev. E, vol. 73, no. 3, (2006), p. 031102. [CrossRef] [Google Scholar]
  14. T. A. M. Langlands, B. I. Henry, S. L. Wearne. Anomalous subdiffusion with multi-species linear reaction dynamics. Phys. Rev. E, vol. 77, no. 2, (2008), p. 021111. [CrossRef] [MathSciNet] [Google Scholar]
  15. S. Fedotov. Non-Markovian random walks and nonlinear reactions: Subdiffusion and propagating fronts. Phys. Rev. E, vol. 81, no. 1, (2010), p. 011117. [CrossRef] [Google Scholar]
  16. E. Abad, S. B. Yuste, K. Lindenberg. Reaction-subdiffusion and reaction-superdiffusion equations for evanescent particles performing continuous-time random walks. Phys. Rev. E, vol. 81, no. 3, (2010), p. 031115. [CrossRef] [Google Scholar]
  17. G. Bel, E. Barkai. Weak ergodicity breaking in the continuous-time random walk. Phys. Rev. Lett., vol. 94, no. 24, (2005), p. 240602. [CrossRef] [Google Scholar]
  18. M. Magdziarz, A. Weron, K. Burnecki, J. Klafter. Fractional Brownian motion versus the continuous-time random walk: A simple test for subdiffusive dynamics. Phys. Rev. Lett., vol. 103, (2009), p. 180602. [CrossRef] [PubMed] [Google Scholar]
  19. W. Deng, E. Barkai. Ergodic properties of fractional Brownian-Langevin motion. Phys. Rev. E, vol. 79, no. 1, (2009), p. 011112. [CrossRef] [MathSciNet] [Google Scholar]
  20. A. Weigel, B. Simon, M. Tamkun, D. Krapf. Ergodic and nonergodic processes coexist in the plasma membrane as observed by single-molecule tracking. Proc. Natl. Acad. Sci., vol. 108, no. 16, (2011), pp. 6438-6443. [CrossRef] [Google Scholar]
  21. T. A. M. Langlands, B. I. Henry. Fractional chemotaxis diffusion equations. Phys. Rev. E, vol. 81, no. 5, (2010), p. 051102. [CrossRef] [MathSciNet] [Google Scholar]
  22. S. Fedotov. Subdiffusion, chemotaxis, and anomalous aggregation. Phys. Rev. E, vol. 83, no. 2, (2011), p. 021110. [CrossRef] [Google Scholar]
  23. I. Eliazar, J. Klafter. Anomalous is ubiquitous. Ann. Phys., vol. 326, no. 9, (2011), pp. 2517-2531. [CrossRef] [Google Scholar]
  24. K. Ritchie, X. Y. Shan, J. Kondo, K. Iwasawa, T. Fujiwara, A. Kusumi. Detection of non-Brownian diffusion in the cell membrane in single molecule tracking. Biophysical journal, vol. 88, no. 3, (2005), p. 2266. [CrossRef] [PubMed] [Google Scholar]
  25. F. Santamaria, S. Wils, E. De Schutter, G. Augustine. The diffusional properties of dendrites depend on the density of dendritic spines. European Journal of Neuroscience, vol. 34, no. 4, (2011), pp. 561-568. [CrossRef] [Google Scholar]
  26. M. Saxton. Anomalous diffusion due to binding: a monte carlo study. Biophysical journal, vol. 70, no. 3, (1996), pp. 1250-1262. [CrossRef] [PubMed] [Google Scholar]
  27. N. Malchus, M. Weiss. Elucidating anomalous protein diffusion in living cells with fluorescence correlation spectroscopy-facts and pitfalls. J. Fluoresc., vol. 20, (2010), pp. 19-26. [CrossRef] [PubMed] [Google Scholar]
  28. J. H. Jeon, V. Tejedor, S. Burov, E. Barkai, C. Selhuber-Unkel, K. Berg-Sørensen, L. Oddershede, R. Metzler, In vivo anomalous diffusion and weak ergodicity breaking of lipid granules. Phys. Rev. Lett., vol. 106, no. 4, (2011), p. 48103. [CrossRef] [PubMed] [Google Scholar]
  29. F. Santamaria, S. Wils, E. De Schutter, G. J. Augustine. Anomalous diffusion in Purkinje cell dendrites caused by spines. Neuron, vol. 52, no. 4, (2006), pp. 635-648. [CrossRef] [PubMed] [Google Scholar]
  30. F. Santamaria, S. Wils, E. De Schutter, G. J. Augustine. The diffusional properties of dendrites depend on the density of dendritic spines. Eur. J. Neurosci., vol. 34, no. 4, (2011), pp. 561-568. [CrossRef] [Google Scholar]
  31. B. I. Henry, T. A. M. Langlands, S. L. Wearne. Fractional cable models for spiny neuronal dendrites. Phys. Rev. Lett., vol. 100, no. 12, (2008), p. 128103. [CrossRef] [PubMed] [Google Scholar]
  32. T. A. M. Langlands, B. I. Henry, S. L. Wearne. Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions J. Math. Biol., vol. 59, no. 6, (2009), pp. 761-808. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  33. T. A. M. Langlands, B. I. Henry, S. L. Wearne. Fractional cable equation models for anomalous electrodiffusion in nerve cells: Finite domain solutions. SIAM J. Appl. Math., vol. 71, no. 4, (2011), pp. 1168-1203. [CrossRef] [Google Scholar]
  34. A. Lubelski J. Klafter. Fluorescence recovery after photobleaching: the case of anomalous diffusion. Biophys. J., vol. 94, no. 12, (2008), pp. 4646-4653. [CrossRef] [PubMed] [Google Scholar]
  35. A. Kolmogoroff, I. Petrovsky, N. Piscounoff. Étude de l’équation de la diffusion avec croissance de la quantité de matiére et son application á un probléme biologique. Moscow Univ. Bull. Math, vol. 1, (1937), pp. 1-25. [Google Scholar]
  36. R. Fisher. The wave of advance of advantageous genes. Ann. Hum. Genet., vol. 7, no. 4, (1937), pp. 355-369. [CrossRef] [Google Scholar]
  37. D. Ben-Avraham, S. Havlin. Diffusion and reactions in fractals and disordered systems. Cambridge University Press, 2000. [Google Scholar]
  38. M. O. Vlad, J. Ross. Systematic derivation of reaction-diffusion equations with distributed delays and relations to fractional reaction-diffusion equations and hyperbolic transport equations: application to the theory of neolithic transition. Phys. Rev. E, vol. 66, no. 6, (2002), p. 061908. [CrossRef] [MathSciNet] [Google Scholar]
  39. C. N. Angstmann, I. C. Donnelly, B. I. Henry. Pattern formation on networks with reactions: A continuous time random walk approach. Phys. Rev. E, vol. 87, no. 3, (2012), p. 032804. [CrossRef] [Google Scholar]
  40. S. Fedotov, S. Falconer. Subdiffusive master equation with space-dependent anomalous exponent and structural insta- bility. Phys. Rev. E, vol 85, no. 3, (2012), p. 031132. [CrossRef] [Google Scholar]
  41. H. Scher, M. Lax. Stochastic transport in a disordered solid. I. Theory. Phys. Rev. B, vol. 7, (1973), pp. 4491-4502. [CrossRef] [MathSciNet] [Google Scholar]
  42. A. Yadav, W. Horsthemke. Kinetic equations for reaction-subdiffusion systems: Derivation and stability analysis. Phys. Rev. E, vol. 74, no. 6, (2006), p. 066118. [CrossRef] [MathSciNet] [Google Scholar]
  43. A. V. Chechkin, R. Gorenflo, I. M. Sokolov. Fractional diffusion in inhomogeneous media. J. Phys. A, vol. 38, (2005), p. L679. [CrossRef] [Google Scholar]
  44. B. Berkowitz, A. Cortis, M. Dentz, H. Scher. Modelling non-Fickian transport in geological formations as a continuous time random walk. Rev. Geophys., vol. 44, (2006), p. RG2003. [CrossRef] [Google Scholar]
  45. E. Scalas, R. Gorenflo, F. Mainardi, M. Raberto. Revisiting the derivation of the fractional diffusion equation. Fractals, vol. 11, (2003), pp. 281-289. [CrossRef] [Google Scholar]
  46. T. H. Hildebrandt. Definitions of Stieltjes integrals of the Riemann type. The Amer. Math. Monthly, vol. 45, (1938), p. 265. [CrossRef] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.